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Univariate ACE Model
Boulder Workshop 2014 Hermine H. Maes, Elizabeth Prom-Wormley
Questions to be Answered
• Does a trait of interest cluster among related individuals?
• Can clustering be explained by genetic or environmental effects?
• What is the best way to explain the degree to which genetic and environmental effects affect a trait?
Practical Example
• Dataset: NH&MRC Twin Register
• 1981 Questionaire
• BMI (body mass index): weight/height squared
• Young Female Cohort: 18-30 years
• Sample Size:
• MZf_young: 534 pairs (zyg=1)
• DZf_young: 328 pairs (zyg=3)
Dataset
!> head(twinData)
fam age zyg part wt1 wt2 ht1 ht2 htwt1 htwt2 bmi1 bmi2
1 115 21 1 2 58 57 1.7000 1.7000 20.0692 19.7232 20.9943 20.8726
2 121 24 1 2 54 53 1.6299 1.6299 20.3244 19.9481 21.0828 20.9519
3 158 21 1 2 55 50 1.6499 1.6799 20.2020 17.7154 21.0405 20.1210
4 172 21 1 2 66 76 1.5698 1.6499 26.7759 27.9155 23.0125 23.3043
5 182 19 1 2 50 48 1.6099 1.6299 19.2894 18.0662 20.7169 20.2583
6 199 26 1 2 60 60 1.5999 1.5698 23.4375 24.3418 22.0804 22.3454
....
Univariate Twin Saturated !
!twinSatConCov.R [1]
# ------------------------------------------------------------------------------
# Program: twinSatConCov.R
# Author: Hermine Maes
# Date: 03 03 2014
#
# Univariate Twin Saturated model to estimate means and (co)variances
# Matrix style model - Raw data - Continuous data
# -------|---------|---------|---------|---------|---------|---------|---------|
!# Load Libraries
require(OpenMx)
require(psych)
!# PREPARE DATA
# Load Data
data(twinData)
dim(twinData)
describe(twinData, skew=F)
Univariate Twin Saturated !
!twinSatConCov.R [2]
# Select Variables for Analysis
Vars <- 'bmi'
nv <- 1 # number of variables
ntv <- nv*2 # number of total variables
selVars <- paste(Vars,c(rep(1,nv),rep(2,nv)),sep="") #c('bmi1','bmi2')
!# Select Covariates for Analysis
twinData[,'age'] <- twinData[,'age']/100
twinData <- twinData[-which(is.na(twinData$age)),]
covVars <- 'age'
!# Select Data for Analysis
mzData <- subset(twinData, zyg==1, c(selVars, covVars))
dzData <- subset(twinData, zyg==3, c(selVars, covVars))
!# Set Starting Values
svMe <- 20 # start value for means
svVa <- .8 # start value for variance
lbVa <- .0001 # start value for lower bounds
svVas <- diag(svVa,ntv,ntv)
lbVas <- diag(lbVa,ntv,ntv)
laMeMZ <- c("m1MZ","m2MZ") # labels for means for MZ twins
laMeDZ <- c("m1DZ","m2DZ") # labels for means for DZ twins
laVaMZ <- c("v1MZ","c21MZ","v2MZ") # labels for (co)variances for MZ twins
laVaDZ <- c("v1DZ","c21DZ","v2DZ") # labels for (co)variances for DZ twins
My Naming Conventions !
!
name of variable(s)
number of variables
number of twin variables
variables per twin pair
number of factors
number of thresholds
!MZ data
DZ data
starting values
lower bound / upper bound
labels
!
Vars <- 'bmi'
nv <- 1
ntv <- nv*2
selVars <-c('bmi1','bmi2')
nf <- 2
nth <- 3
!mzData
dzData
sv
lb / ub
la
Classical Twin Study Background
• The Classical Twin Study (CTS) uses MZ and DZ twins reared together
• MZ twins share 100% of their genes
• DZ twins share on average 50% of their genes
• Expectation: Genetic factors are assumed to contribute to a phenotype when MZ twins are more similar than DZ twins
Classical Twin Study Assumptions
• Equal Environments of MZ and DZ pairs
• Random Mating
• No GE Correlation
• No G x E Interaction
• No Sex Limitation
• No G x age Interaction
Classical Twin Study Basic Data Assumptions• MZ and DZ twins are sampled from the
same population, therefore we expect :
• Equal means/variances in Twin 1 and Twin 2
• Equal means/variances in MZ and DZ twins
• Further assumptions would need to be tested if we introduce male twins and opposite sex twin pairs
‘Old Fashioned’ Data Checking
MZ DZ
T1 T2 T1 T2
mean 21.35 21.34 21.45 21.46
variance 0.73 0.79 0.77 0.82
covariance 0.59 0.25
!
!
!
!
!
Nice, but how can we actually be sure that these means and variances are truly the same?
Univariate Analysis A Roadmap
1. Use data to test basic assumptions (equal means & variances for twin 1/twin 2 and MZ/DZ pairs)
• Saturated Model
2. Estimate contributions of genetic/environmental effects on total variance of a phenotype
• ACE or ADE Models
3. Test ACE (ADE) submodels to identify and report significant genetic and environmental contributions
• AE or CE or E Only Models
Probability Density Function Φ(xi)
Φ(xi) is likelihood of data point xi for particular mean and variance estimates !
Φ(xi)= -|2πσ2|-.5 e -.5((xi - µ)2 /σ2) π: pi=3.14; xi: observed value of variable i; µ: expected mean; σ: expected variance
Univariate: height of probability density function
Multinormal Probability Function
Φ(xi) is likelihood of pair of data points xi and yi for particular means, variances and correlation estimates !Φ(xi)= -|2π∑|-n/2 e -.5((xi-µ)∑-1(xi-µ)’) π=3.14; xi: value of variable i; µ: expected mean; ∑: expected covariance matrix
Multivariate: height of multinormal probability density function
rMZ=.85
rMZ=.85
rDZ=.49
Intuition behind Maximum Likelihood (ML)• Likelihood: probability that an observation (data point) is
predicted by specified model
• For MLE, determine most likely values of population parameter value (e.g, µ, σ, β) given observed sample value
• define model
• define probability of observing a given event conditional on a particular set of parameters
• choose a set of parameters which are most likely to have produced observed results
Likelihood Ratio Test
• Likelihood Ratio test is a simple comparison of Log Likelihoods under 2 separate models:
• Model Mu is Unconstrained (has more parameters)
• Model Mc is Constrained (has fewer parameters)
• LR statistic equals:
• LR (Mc | Mu) = 2ln(L(Mu) - 2ln(L(Mc)
• LR is asymptotically distributed as χ2 with the df equal to the number of constraints
Predicted MeansT1 T2
m1MZ +b*Age m2MZ +b*Age
1x2 matrices
1T1 μ1MZ T2μ2MZ
var1MZ
covMZ
MZ var2MZ
b AgeAge
defAge <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE, labels=c("data.age"), name=“Age" ) pathB <- mxMatrix( type="Full", nrow=1, ncol=1, free=TRUE, values= .01, label="l11", name="b" ) laMeMZ <- c(“m1MZ”,"m2MZ"); laMeDZ <- c(“m1DZ","m2DZ") !meanMZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=svMe, labels=laMeMZ, name="MeanMZ") meanDZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=svMe, labels=laMeDZ, name="MeanDZ") expMeanMZ <- mxAlgebra( expression= meanMZ + cbind(b%*%Age,b%*%Age), name="expMeanMZ" ) expMeanDZ <- mxAlgebra( expression= meanDZ + cbind(b%*%Age,b%*%Age), name="expMeanDZ" )
T1 T2m1DZ +b*Age m2DZ +b*Age
1T1 μ1DZ T2μ2DZ
var1DZ
covDZ
var2DZ
b AgeAge
Predicted CovariancesT1 T2
T1 v1MZ c21MZ
T2 c21MZ v2MZ
2x2 matrices
T1 T2
T1 v1DZ c21DZ
T2 c21DZ v2DZ
laVaMZ <- c("v1MZ","c21MZ","v2MZ") laVaDZ <- c(“v1DZ","c21DZ","v2DZ") !covMZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=svVas, lbound=lbVas, labels=laVaMZ, name="expCovMZ" ) covDZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE, values=svVas, lbound=lbVas, labels=laVaDZ, name="expCovDZ" )
1T1 μ1MZ T2μ2MZ
v1MZ
c21MZ
v2MZ
b AgeAge
1T1 μ1DZ T2μ2DZ
v1DZ
c21DZ
v2DZ
b AgeAge
Univariate Twin Saturated !
!twinSatCon.R [3]
# Matrices for Covariates and linear Regression Coefficients
defAge <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE,
labels=c("data.age"), name="Age")
pathB <- mxMatrix( type="Full", nrow=1, ncol=1, free=TRUE,
values=.01, label="l11", name="b")
!# Algebra for expected Mean Matrices in MZ & DZ twins
meanMZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=svMe,
labels=laMeMZ, name="meanMZ" )
meanDZ <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=svMe,
labels=laMeDZ, name="meanDZ" )
expMeanMZ <- mxAlgebra( expression= meanMZ + cbind(b%*%Age,b%*%Age), name="expMeanMZ" )
expMeanDZ <- mxAlgebra( expression= meanDZ + cbind(b%*%Age,b%*%Age), name="expMeanDZ" )
!# Algebra for expected Variance/Covariance Matrices in MZ & DZ twins
expCovMZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE,
values=svVas, lbound=lbVas, labels=laVaMZ, name="expCovMZ" )
expCovDZ <- mxMatrix( type="Symm", nrow=ntv, ncol=ntv, free=TRUE,
values=svVas, lbound=lbVas, labels=laVaDZ, name="expCovDZ" )
!
Univariate Twin Saturated !
!twinSatCon.R [4]
# Data objects for Multiple Groups
dataMZ <- mxData( observed=mzData, type="raw" )
dataDZ <- mxData( observed=dzData, type="raw" )
!# Objective objects for Multiple Groups
objMZ <- mxFIMLObjective( covariance="expCovMZ", means=“expMeanMZ”,
dimnames=selVars )
objDZ <- mxFIMLObjective( covariance="expCovDZ", means=“expMeanDZ”,
dimnames=selVars )
!# Combine Groups
modelMZ <- mxModel( "MZ", defAge, pathB, meanMZ, expMeanMZ, expCovMZ, dataMZ,
objMZ )
modelDZ <- mxModel( "DZ", defAge, pathB, meanDZ, expMeanDZ, expCovDZ, dataDZ,
objDZ )
minus2ll <- mxAlgebra( MZ.objective+ DZ.objective, name="minus2sumloglikelihood" )
obj <- mxFitFunctionAlgebra( "minus2sumloglikelihood" )
ciCov <- mxCI( c('MZ.expCovMZ','DZ.expCovDZ' ))
ciMean <- mxCI( c('MZ.expMeanMZ','DZ.expMeanDZ' ))
twinSatModel <- mxModel( "twinSat", modelMZ, modelDZ, minus2ll, obj,
ciCov, ciMean )
Univariate Twin Saturated !
!twinSatCon.R [5]
# ------------------------------------------------------------------------------
# RUN MODEL
!# Run Saturated Model
twinSatFit <- mxRun( twinSatModel, intervals=F )
twinSatSum <- summary( twinSatFit )
twinSatSum
!# Generate Saturated Model Output
twinSatFit$MZ.expMeanMZ@result
twinSatFit$DZ.expMeanDZ@result
twinSatFit$MZ.expCovMZ@values
twinSatFit$DZ.expCovDZ@values
!twinSatSum$observedStatistics
length(twinSatSum$parameters[[1]])
twinSatFit@output$Minus2LogLikelihood
twinSatSum$degreesOfFreedom
twinSatSum$AIC
round(twinSatFit@output$estimate,4)
summary(MxModel) !!
free parameters: name matrix row col Estimate Std.Error lbound 1 l11 MZ.b 1 1 2.7536831 0.69736577 2 mMZ1 MZ.meanMZ 1 1 20.6886948 0.16989840 3 mMZ2 MZ.meanMZ 1 2 20.6934721 0.17017616 4 vMZ1 MZ.expCovMZ bmi1 bmi1 0.7213571 0.04326293 1e-04 5 cMZ21 MZ.expCovMZ bmi1 bmi2 0.5840681 0.04034558 0 6 vMZ2 MZ.expCovMZ bmi2 bmi2 0.7842634 0.04713262 1e-04 7 mDZ1 DZ.meanDZ 1 1 20.7833537 0.17217692 8 mDZ2 DZ.meanDZ 1 2 20.8080578 0.17278599 9 vDZ1 DZ.expCovDZ bmi1 bmi1 0.7281361 0.05594823 1e-04 10 cDZ21 DZ.expCovDZ bmi1 bmi2 0.2414624 0.04378434 0 11 vDZ2 DZ.expCovDZ bmi2 bmi2 0.8030416 0.06183128 1e-04 !observed statistics: 1775 estimated parameters: 11 degrees of freedom: 1764 -2 log likelihood: 4015.118 number of observations: 919 Information Criteria AIC: 1837.118
Estimated Values
T1 T2 T1 T2Saturated Model
mean MZ 20.68 20.69 DZ 20.78 20.80
cov T1 0.72 T1 0.73
T2 0.58 0.78 T2 0.24 0.80
11 parameters Estimated: l11 m1MZ, m2MZ, v1MZ, v2MZ, c21MZ m1DZ, m2DZ, v1DZ, v2DZ, c21DZ
Goodness-of-Fit Statistics
os observed statistics
ep estimated parameters
-2ll -2 LogLikelihood
df degrees of freedom os - ep
AIC Akaike’s Information Criterion -2ll -2df
diff -2ll likelihood ratio Chi-square
diff df difference in degrees of freedom
ep -2ll df AICdiff -2ll
diff df p
Sat 11 4015.12 1764 487.12
Fitting Submodels !!
# Test significance of Covariate !# Copy model, provide new name testCovModel <- mxModel(twinSatFit, name=“testCov”) !# Change parameter by changing attributes for label testCovModel <- omxSetParameters( testCovModel, label=“l11", free=FALSE, values=0 ) !# Fit Nested Model testCovFit <- mxRun(testCovModel) !# Compare Nested Model with ‘Full’ Model mxCompare(twinSatFit, testCovFit)
Goodness-of-Fit Stats
ep -2ll df AIC diff -2ll
diff df
p
Saturated 11 4015.12 1764 487.12
drop beta 10 4030.57 1765 500.57 15.45 1 0
Fitting Nested Models
• Saturated Model • likelihood of data without any constraints
• fitting as many means and (co)variances as possible
• Equality of means & variances by twin order • test if mean of twin 1 = mean of twin 2
• test if variance of twin 1 = variance of twin 2
• Equality of means & variances by zygosity • test if mean of MZ = mean of DZ
• test if variance of MZ = variance of DZ
Equate Means across twin order
twinSatCon.R [7]# ------------------------------------------------------------------------------
# RUN SUBMODELS
!# Constrain expected Means to be equal across twin order
eqMeansTwinModel <- mxModel(twinSatFit, name="eqMeansTwin" )
eqMeansTwinModel <- omxSetParameters( eqMeansTwinModel, label=c("m1MZ","m2MZ"), free=TRUE, values=svMe, newlabels='mMZ' ) eqMeansTwinModel <- omxSetParameters( eqMeansTwinModel, label=c("m1DZ","m2DZ"), free=TRUE, values=svMe, newlabels='mDZ' ) !eqMeansTwinFit <- mxRun( eqMeansTwinModel, intervals=F )
eqMeansTwinSum <- summary( eqMeansTwinFit )
eqMeansTwinLLL <- eqMeansTwinFit@output$Minus2LogLikelihood
!twinSatLLL <- twinSatFit@output$Minus2LogLikelihood
chi2Sat_eqM <- eqMeansTwinLLL-twinSatLLL
pSat_eqM <- pchisq( chi2Sat_eqM, lower.tail=F, 2)
chi2Sat_eqM; pSat_eqM
mxCompare(twinSatFit, eqMeansTwinFit)
Equate Means & Variances across twin order & zygosity
!twinSatCon.R [8]
# Constrain expected Means and Variances to be equal across twin order
eqMVarsTwinModel <- mxModel(eqMeansTwinFit, name="eqMVarsTwin" )
eqMVarsTwinModel <- omxSetParameters( eqMVarsTwinModel, label=c("v1MZ","v2MZ"), free=TRUE, values=svMe, newlabels='vMZ' ) eqMVarsTwinModel <- omxSetParameters( eqMVarsTwinModel, label=c("v1DZ","v2DZ"), free=TRUE, values=svMe, newlabels='vDZ' ) eqMVarsTwinFit <- mxRun( eqMVarsTwinModel, intervals=F )
subs <- list(eqMeansTwinFit, eqMVarsTwinFit)
mxCompare(twinSatFit, subs)
!# Constrain expected Means and Variances to be equal across order and zygosity
eqMVarsZygModel <- mxModel(eqMVarsTwinModel, name="eqMVarsZyg" )
eqMVarsZygModel <- omxSetParameters( eqMVarsZygModel,
label=c("mMZ","mDZ"), free=TRUE, values=svMe, newlabels='mZ' )
eqMVarsZygModel <- omxSetParameters( eqMVarsZygModel,
label=c("vMZ","vDZ"), free=TRUE, values=svMe, newlabels='vZ' )
eqMVarsZygFit <- mxRun( eqMVarsZygModel, intervals=F )
mxCompare(eqMVarsTwinFit, eqMVarsZygFit)
Estimated Values
T1 T2 T1 T2Equate Means & Variances across Twin Order
mean MZ DZcov T1 T1
T2 T2Equate Means Variances across Twin Order & Zygositymean MZ DZcov T1 T1
T2 T2
Goodness-of-Fit Stats
ep -2ll df AIC diff -2ll
diff df
p
Saturated
mT1=mT2
mT1=mT2 varT1=varT2
Zyg MZ=DZ
Goodness-of-Fit Stats
ep -2ll df AIC diff -2ll
diff df
p
Saturated 11 4015.12 1764 487.12
mT1=mT2 9 4015.35 1766 483.35 0.23 2 0.89
mT1=mT2 varT1=varT2 7 4018.61 1768 482.61 3.49 4 0.48
Zyg MZ=DZ 5 4022.79 1779 482.78 7.67 6 0.26
Patterns of Twin Correlations
rMZ = 2rDZ!Additive!!!
DZ twins !on average
share 50% of additive!effects!
rMZ = rDZ!Shared
Environment!!
rMZ > 2rDZ!Additive &!Dominance!
!DZ twins!
on average share 25% of
dominance effects!
A = 2(rMZ-rDZ)!C = 2rDZ - rMZ! E = 1- rMZ!
rDZ > ½ rMZ!Additive &
Shared Environment!
Twin Correlations ~ Sources of Variance
1-rMZ
rMZ > rDZ
rMZ =2 rDZ
rMZ = rDZ
rMZ < 1/2 rDZ
rMZ > 1/2 rDZ
E
A
only A
only C
A & C
A & D
Twin Correlations
18 20 22 24 26
1820
2224
26
MZ BMI
MZ bmi1
MZ
bmi2
Corr = 0.78
18 20 22 24 26
1820
2224
26
DZ BMI
DZ bmi1
DZ
bmi2
Corr = 0.30
Univariate ADE / ACE Model
1
1
T1
e11
μ T2
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1
1
μ
MZ
1
1
T1
e11
μ T2
E1
1
C1
1
A1
1
A2
1
C2
1
E2
c11 a11 e11a11 c11
1
1
μ
MZ
1
1
T1
e11
μ T2
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
0.5
0.25
μ
DZ
1
1
T1
e11
μ T2
E1
1
C1
1
A1
1
A2
1
C2
1
E2
c11 a11 e11a11 c11
0.5
1
μ
DZ
Univariate Analysis A Roadmap
1. Use data to test basic assumptions (equal means & variances for twin 1/twin 2 and MZ/DZ pairs)
• Saturated Model
2. Estimate contributions of genetic/environmental effects on total variance of a phenotype
• ACE or ADE Models
3. Test ACE (ADE) submodels to identify and report significant genetic and environmental contributions
• AE or CE or E Only Models
Paths & Variance Components !
!twinAdeCon.R [3]
# ------------------------------------------------------------------------------
# PREPARE MODEL
!# Set Starting Values
svMe <- 20 # start value for means
svPa <- .6 # start value for path coefficients
(sqrt(variance/#ofpaths))
!# ADE Model
# Matrices declared to store a, d, and e Path Coefficients
pathA <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa,
label="a11", name="a" )
pathD <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa,
label="d11", name="d" )
pathE <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa,
label="e11", name="e" )
# Matrices generated to hold A, D, and E computed Variance Components
covA <- mxAlgebra( expression=a %*% t(a), name="A" )
covD <- mxAlgebra( expression=d %*% t(d), name="D" )
covE <- mxAlgebra( expression=e %*% t(e), name="E" )
ADE Deconstructed Path Coefficients
a 1 x 1
a11
d 1 x 1
d11
e 1 x 1
e11
1
1
T1
e11
T2
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.25
1 or 0.5
pathA <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa, label="a11", name="a" ) !pathD <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa, label="d11", name="d" ) !pathE <- mxMatrix( type="Full", nrow=nv, ncol=nv, free=TRUE, values=svPa, label="e11", name="e" )
ADE Deconstructed Variance Components
A 1 x 1
a11
D 1 x 1
d11
E 1 x 1
e11
a11T
d11T
e11T
*
*
*
1
1
T1
e11
T2
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.25
1 or 0.5
covA <- mxAlgebra( expression=a %*% t(a), name="A" ) !covD <- mxAlgebra( expression=d %*% t(d), name="D" ) !covE <- mxAlgebra( expression=e %*% t(e), name="E" )
Expected Means & (Co)Variances
twinAdeCon.R [4]
# Matrices for covariates and linear regression coefficients
defAge <- mxMatrix( type="Full", nrow=1, ncol=1, free=FALSE,
labels=c("data.age"), name="Age")
pathB <- mxMatrix( type="Full", nrow=1, ncol=1, free=TRUE,
values= .01, label="l11", name="b" )
!# Algebra for expected Mean Matrices in MZ & DZ twins
meanG <- mxMatrix( type="Full", nrow=1, ncol=ntv, free=TRUE, values=svMe,
labels="xbmi", name="mean" )
expMean <- mxAlgebra( expression= mean + cbind(b%*%Age,b%*%Age),
name="expMean" )
!# Algebra for expected Variance/Covariance Matrices in MZ & DZ twins
covP <- mxAlgebra( expression= A+D+E, name="V" )
expCovMZ <- mxAlgebra( expression= rbind( cbind(V, A+D), cbind(A+D, V)),
name="expCovMZ" )
expCovDZ <- mxAlgebra( expression= rbind( cbind(V, 0.5%x%A+ 0.25%x%D),
cbind(0.5%x%A+ 0.25%x%D , V)), name="expCovDZ" )
ADE Deconstructed Means & Variances
expMean 1 x 2
mean mean
expCovMZ 2 x 2
VV
expCovDZ 2 x 2
1
1
T1
e11
mean T2mean
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.25
1 or 0.5
covP <- mxAlgebra( expression= A+D+E, name="V" ) !covMZ <- mxAlgebra( expression= rbind( cbind(V, A+D), cbind(A+D, V)), name="expCovMZ") !covDZ <- mxAlgebra( expression= rbind( cbind(V, 0.5%x%A+ 0.25%x%D), cbind(0.5%x%A+ 0.25%x%D, V)), name="expCovDZ")
ADE Deconstructed A Covariances
expCovMZ 2 x 2
V A+D
A+D V
expCovDZ 2 x 2
V .5A+.25D
.5A+.25D V
1
1
T1
e11
mean T2mean
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.25
1 or 0.5
covP <- mxAlgebra( expression= A+D+E, name="V" ) !covMZ <- mxAlgebra( expression= rbind( cbind(V, A+D), cbind(A+D, V)), name="expCovMZ") !covDZ <- mxAlgebra( expression= rbind( cbind(V, 0.5%x%A+ 0.25%x%D), cbind(0.5%x%A+ 0.25%x%D, V)), name="expCovDZ")
ADE Deconstructed D Covariances
expCovMZ 2 x 2
expCovDZ 2 x 2
V A+D
A+D V
V .5A+.25D
.5A+.25D V
1
1
T1
e11
mean T2mean
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.5
1 or 0.25
covP <- mxAlgebra( expression= A+D+E, name="V" ) !covMZ <- mxAlgebra( expression= rbind( cbind(V, A+D), cbind(A+D, V)), name="expCovMZ") !covDZ <- mxAlgebra( expression= rbind( cbind(V, 0.5%x%A+ 0.25%x%D), cbind(0.5%x%A+ 0.25%x%D, V)), name="expCovDZ")
ADE Deconstructed Parameters
!!!!!!!!5 Parameters Estimated: Mean xbmi Regression on Age b11 Variance due to A a11 Variance due to D d11 Variance due to E e11
1
1
T1
e11
xbmi T2
E1
1
D1
1
A1
1
A2
1
D2
1
E2
d11 a11 e11a11 d11
1 or 0.5
1 or 0.25
xbmi
b11 AgeAge
Data & Objectives !
!twinAdeCon.R [5]
# Data objects for Multiple Groups
dataMZ <- mxData( observed=mzData, type="raw" )
dataDZ <- mxData( observed=dzData, type="raw" )
!# Objective objects for Multiple Groups
objMZ <- mxFIMLObjective( covariance="expCovMZ", means=“expMean",
dimnames=selVars )
objDZ <- mxFIMLObjective( covariance="expCovDZ", means=“expMean",
dimnames=selVars )
!# Combine Groups
pars <- list( pathA, pathD, pathE, covA, covD, covE, covP, pathB )
modelMZ <- mxModel( pars, defAge, meanG, expMean, expCovMZ, dataMZ, objMZ,
name="MZ" )
modelDZ <- mxModel( pars, defAge, meanG, expMean, expCovDZ, dataDZ, objDZ,
name="DZ" )
minus2ll <- mxAlgebra( expression=MZ.objective + DZ.objective, name="m2LL" )
obj <- mxAlgebraObjective( "m2LL" )
AdeModel <- mxModel( "ADE", pars, modelMZ, modelDZ, minus2ll, obj )
Model Fitting !
!twinAdeCon.R [6]
# RUN MODEL
!# Run ADE model
AdeFit <- mxRun(AdeModel, intervals=T)
AdeSumm <- summary(AdeFit)
AdeSumm
mxCompare(twinSatFit,AdeFit)
round(AdeFit@output$estimate,4)
round(AdeFit$Vars@result,4)
!# Generate Table of Parameter Estimates using mxEval
pathEstimatesADE <- print(round(mxEval(cbind(a,d,e), AdeFit),4))
varComponentsADE <- print(round(mxEval(cbind(A/V,D/V,E/V), AdeFit),4))
rownames(pathEstimatesADE) <- 'pathEstimates'
colnames(pathEstimatesADE) <- c('a','d','e')
rownames(varComponentsADE) <- 'varComponents'
colnames(varComponentsADE) <- c('a^2','d^2','e^2')
pathEstimatesADE
varComponentsADE
Generating Output !
!twinAdeCon.R [7]
# Generate ADE Model Output
estMean <- mxEval(expMean, AdeFit$MZ) # expected mean
estCovMZ <- mxEval(expCovMZ, AdeFit$MZ) # expected covariance matrix for MZ's
estCovDZ <- mxEval(expCovDZ, AdeFit$DZ) # expected covariance matrix for DZ's
estVA <- mxEval(a*a, AdeFit) # additive genetic variance, a^2
estVD <- mxEval(d*d, AdeFit) # dominance variance, d^2
estVE <- mxEval(e*e, AdeFit) # unique environmental variance, e^2
estVP <- (estVA+estVD+estVE) # total variance
estPropVA <- estVA/estVP # standardized additive genetic variance
estPropVD <- estVD/estVP # standardized dominance variance
estPropVE <- estVE/estVP # standardized unique environmental var
estADE <- rbind(cbind(estVA,estVD,estVE), # table of estimates
cbind(estPropVA,estPropVD,estPropVE))
LL_ADE <- mxEval(objective, AdeFit) # likelihood of ADE model
!
summary(mxModel) !
!
free parameters: name matrix row col Estimate Std.Error lbound ubound 1 a11 a 1 1 0.6060900 NaN 2 d11 d 1 1 0.4743898 NaN 3 e11 e 1 1 0.4111268 NaN 4 l11 b 1 1 2.7677606 NaN 5 xbmi MZ.mean 1 1 20.7346094 NaN !confidence intervals: lbound estimate ubound ADE.Vars[1,1] 6.811043e-02 0.3673451 0.6333987 ADE.Vars[1,2] 5.966440e-21 0.2250457 0.5243742 ADE.Vars[1,3] 1.502990e-01 0.1690253 0.1909726 ADE.Vars[1,4] 8.988222e-02 0.4824499 0.7934488 ADE.Vars[1,5] 1.415784e-13 0.2955620 0.6894092 ADE.Vars[1,6] 1.937827e-01 0.2219881 0.2545650 !observed statistics: 1775 estimated parameters: 5 degrees of freedom: 1770 -2 log likelihood: 4022.789 number of observations: 919 Information Criteria AIC: 1832.789
Goodness-of-Fit Stats
ep -2ll df AIC diff -2ll
diff df
p
Saturated 11 4015.12 1764 487.12
ADE 5 4022.79 1770 482.79 7.67 6 0.26
Table of Estimates !
!
> # Generate Table of Parameter Estimates using mxEval > pathEstimatesADE a d e pathEstimates 0.6061 0.4744 0.4111 > varComponentsADE a^2 d^2 e^2 varComponents 0.4824 0.2956 0.222
Univariate Analysis A Roadmap
1. Use data to test basic assumptions (equal means & variances for twin 1/twin 2 and MZ/DZ pairs)
• Saturated Model
2. Estimate contributions of genetic/environmental effects on total variance of a phenotype
• ACE or ADE Models
3. Test ACE (ADE) submodels to identify and report significant genetic and environmental contributions
• AE or CE or E Only Models
Nested Models
• ‘Full’ ADE Model • Nested Models • AE Model • test significance of D
• E Model vs AE Model • test significance of A
• E Model vs ADE Model • test combined significance of A & D
AE Deconstructed Parameters
1
1
T1
e11
mean T2mean
E1
1
A1
1
A2
1
E2
a11 e11a11
1 or 0.5
# Test significance of D # Copy model, provide new name AeModel <- mxModel(AdeFit, name=“AE”) !# Change parameter by changing attributes for label AeModel <- omxSetParameters( AeModel, label=“d11", free=FALSE, values=0 ) !# Fit Nested Model AeFit <- mxRun(AeModel) !# Compare Nested Model with ‘Full’ Model mxCompare(AdeFit, AeFit)
AE Model !
!twinAdeCon.R [8]
# ------------------------------------------------------------------------------
# FIT SUBMODELS
!# Run AE model
AeModel <- mxModel( AdeFit, name="AE" )
AeModel <- omxSetParameters( AeModel, labels="d11", free=FALSE, values=0 )
AeFit <- mxRun(AeModel)
mxCompare(AdeFit, AeFit)
round(AeFit@output$estimate,4)
round(AeFit$Vars@result,4)
!# Run E model
eModel <- mxModel( AeFit, name="E" )
eModel <- omxSetParameters( eModel, labels="a11", free=FALSE, values=0 )
eFit <- mxRun(eModel)
mxCompare(AeFit, eFit)
round(eFit@output$estimate,4)
round(eFit$Vars@result,4)
!# Print Comparative Fit Statistics
AdeNested <- list(AeFit, eFit)
mxCompare(AdeFit,AdeNested)
round(rbind(AdeFit$Vars@result,AeFit$Vars@result,eFit$Vars@result),4)
Goodness-of-Fit Statistics
ep -2ll df AICdiff -2ll
diff df p
ADE
AE
E
Estimated Values
path coefficients
unstandardized variance
components
standardized variance
components
a d e a2 d2 e2 a2 d2 e2
ADE
AE
E
Goodness-of-Fit Statistics
ep -2ll df AICdiff -2ll
diff df p
ADE 5 4022.79 1770 482.79
AE 4 4025.41 1771 483.41 2.62 1 0.10
E 3 4549.61 1772 1005.61 526.8 2 0
Estimated Values
path coefficients
unstandardized variance
components
standardized variance
components
a d e a2 d2 e2 a2 d2 e2
ADE 0.61 0.47 0.41 0.37 0.25 0.17 0.48 0.30 0.22
AE 0.77 - 0.41 0.60 - 0.17 0.77 - 0.22
E - - 0.87 - - 0.76 - - 1.00
What about C?
• ‘Full’ ACE Model • Nested Models • AE Model • test significance of C
• CE Model • test significance of A
• E Model vs AE Model • test significance of A
• E Model vs ACE Model • test combined significance of A & C
Goodness-of-Fit Statistics
ep -2ll df AICdiff -2ll
diff df p
ADE
AE
ACE
CE
E
Estimated Values
a d e c a2 d2 e2 c2
ADE
AE
ACE
AE
E
Conclusions
• BMI in young OZ females (age 18-30)
• additive genetic factors: highly significant
• dominance: borderline non-significant
• specific environmental factors: significant
• shared environmental factors: not
Recommended